Yes! Microchip! Great idea, Daisycar.
I had a situation that wasn't the same, but kind of similar. Years ago, we bought a property directly from the owner, and there were two dogs left there. The ex-owners told us the dogs belonged to a relative who was coming to pick them up in a couple weeks when he had time off, and in the meantime, "just throw some food under the porch, and they can drink from the pond." The relative did show up on schedule, but he only took the black lab and said the golden retriever wasn't his.
So we kept the retriever - he was a really nice older dog and neutered, so somebody must have cared about him at some point. After we talked to the neighbors, they said the dog's name was Pete and he'd belonged to the boy who'd lived there two owners ago. The neighbor's teenage daughter had played with the boy (and Pete) when they were little kids. We got all Pete's vet care done and he became a much-loved member of the family.
After few years, he was diagnosed with cancer and radiographs showed it had spread, so we just kept him as comfortable as possible and spoiled him. When Pete's time was coming close to the end, I wanted to somehow let his ex-boy know what was happening and give him a chance to say goodbye if he wanted to - he had obviously loved Pete in the past and it's not a child's fault when their parents choose to abandon a dog.
I got their name from the neighbors and did some research to get the mother's phone number. When I called her, she sounded really uncaring, told me her son was currently incarcerated at the juvenile detention center, and tried to talk me into taking her current unwanted dog.
Oh well, I tried...when Pete was ready to go, the vet came out to our place instead of having to take him in, and we buried him under a tree near the pond where he spent so many happy times.
Anyway, FeatherLace, Bear is yours now. Love the pictures, she's a cutie! And her shiny coat is a testament to the great care you're giving her. You are doing the right thing, and your neighbors can take a long walk off a short pier, they sound like jerks, frankly. I really don't understand why some people even get dogs, when they don't either want them as family members or train them to do a specific job.